On Coefficient problems for classes ${\mathcal{S}}_e^{\ast }$ and ${\mathcal{C}}_e$

Logarithmic coefficients play a crucial role in the theory of univalent functions. In this study,we focus on the classes ${\mathcal{S}}_e^{\ast }$ and ${\mathcal{C}}_e$ of starlike and convex functions, respectively, \begin{align*} $\mathcal{S}$_e^\ast := \left{ f \in $\mathcal{S}$ : \frac{zf’(z)}{f(z)} \prec e^z, \ z \in $\mathbb{D}$ \right}, \end{align*} and \begin{align*} $\mathcal{C}$_e := \left{ f \in $\mathcal{S}$ : 1 + \frac{z f’‘(z)}{f’(z)} \prec e^z, \ z \in $\mathbb{D}$ \right}. \end{align*} This paper investigates the sharp bounds of the logarithmic coefficients and the Hermitian-Toeplitz determinant of these coefficients for the classes ${\mathcal{S}}_e^{\ast }$ and ${\mathcal{C}}_e$. Additionally, we examine the generalized Zalcman conjecture and the generalized Fekete-Szegö inequality for these classes ${\mathcal{S}}_e^{\ast }$ and ${\mathcal{C}}_e$ and show that the inequalities are sharp.